This invention is concerned with an improved method for transforming a set of spatially-distributed non-zero-offset seismic traces into a new set of dip-resolved, zero-offset, unmigrated output seismic traces representative of the three-dimensional (3-D) configuration of a volume of the earth.
Common Mid Point (CMP) stacking, also sometimes referred to as Common Depth Point or Common Reflection Point, CDP or CRP respectively) of seismic field data is well known. See for example U.S. Pat. Nos. 3,217,828 to Mendenhall et al, and 2,732,906 to Mayne, which are incorporated herein by reference as a teaching of the CMP technique. The U.S. Pat. No. 3,217,828 teaches two-dimensional (2-D) data processing where dipping earth layers are projected into a two-dimensional plane, perpendicular to the surface of the earth, along a designated line of profile.
The earth is three-dimensional. It is the configuration of a volume of the earth that is of interest for mapping three-dimensional structures likely to contain valuable mineral resources. In 2-D processing, only the dip component parallel to the line of survey is mapped. Earth-layer dip components lying outside the 2-D plane are distorted.
The limitations of conventional 2-D calculations are three-fold: First, for a dipping reflector, the true reflecting point for a reflection at a non-zero offset lies up-dip from the midpoint between a seismic source and a receiver of seismic signals. When the data from different offsets are stacked, reflection-point smear occurs.
Second, stacking velocities are dip-dependent and hence when events with differing dips cross on a processed seismic section, no single velocity function will properly align the different events on the CMP gather at that location. That situation causes conventional stacking to mid-stack the events for which the chosen velocity was inappropriate.
Third, the stacking velocities are azimuth-dependent. That is, the stacking velocity depends on the orientation of the direction of maximum dip with respect to the vertical plane between the seismic source and the receiver.
Because 2-D exploration is confined to single lines of profile, 3-D resolution of the details of a volume of the earth based upon 2-D is often somewhat distorted.
3-D areal coverage often takes the form of a grid of seismic lines consisting essentially of two sets of lines orthogonally arranged with respect to one another. Each line includes a plurality of seismic-signal receivers interspersed with a plurality of seismic sources that are placed at desired locations within the grid. The grid may, however, be rectangular, circular, hexagonal, star-shaped or any other desired geometrical pattern. The grid of sources and receivers is customarily placed horizontally over the surface of the earth but conceptually at least, there is no reason why vertical arrays should be excluded. With suitable equipment adaptations, 3-D operations may be carried out on land or at sea.
Within a reasonable radius of about two miles or so, the firing of any one source will isonify an entire array of seismic sensors or receivers in a large area. Accordingly, any source-receiver pair, or combination thereof, along any desired azimuth therebetween, may be designated as a "line of profile".
The area of survey may be divided into cells of suitable geometrical shape such as a rectangle. The cell of dimensions depend upon the ultimate resolution desired but dimensions such as 25 by 50 meters or less are not uncommon. In processing, for any given source-receiver pair, there may be designated a midpoint; the midpoint is assigned to an appropriate cell for mapping and other purposes.
U.S. Pat. Nos. 4,241,429; 4,330,873; and 4,403,312 are typical of prior-art 3-D seismic exploration arrays. F. K. Levin in "Apparent Velocity from Dipping Interface Reflections," Geophysics, 1971, pp. 510-516, discusses the dependence of the medium velocity on dip and azimuth.
Ordinarily, the geophysicist has no prior knowledge of local dip or azimuth of a line of profile with respect to the direction of maximum dip. There is therefore always the problem of selection of the prior migration velocity function for dip migration. Yet that problem is of key interest when steep dips, substantially greater than about 10 degrees, are present. As Levin pointed out, and as will be discussed later, if the migration velocity is not corrected for both dip and azimuth when reflection-point data derived from different offsets and azimuths are stacked, the data are smeared.
The most direct method of overcoming reflection-point smear is to perform migration before stack. That technique converts non-zero offset data directly to the final migrated image without any restriction on dip, offset or azimuth. Full migration before stack is, however, a very costly process computationally because, for a 3-D survey, as many as 2 to 10 million traces, each including 1500 to 2000 samples, must be individually processed. Furthermore, the migration velocity field must be known with considerable accuracy before pre-stack migration is possible. That requirement involves additional expensive velocity analyses. It is evident therefore that the data-compression capability offered by stacking before migration is highly desirable.
An attractive alternative to full migration before stack in 2-D is the use of Pre-stack Partial Migration, also known as Dip Moveout Correction (DMO). That process attempts the relatively modest task of converting finite-offset data that have had a conventional normal moveout correction (NMO) applied using migration velocities, to true zero offset. DMO removes the effect of reflection-point smear and enables events to be stacked coherently regardless of dip or azimuth. Thereafter the data can be stacked and migrated conventionally but with a substantial degree of data compression. An additional advantage is that DMO is a relatively small correction that tends to be insensitive to errors in the estimated velocity used for NMO correction.
Hale proposed a DMO algorithm for 2-D data reduction in his Doctoral thesis "Dip Movement by Fourier Transform", submitted to the Department of Geophysics, Stanford University, May, 1983. On pages 67-71, he proposes to extend his 2-D analysis from 2-D to 3-D but fails to arrive at a computationally practical analytical solution.
A major disadvantage of the Hale transform in 3-D is that it cannot be directly reformulated as a fourier transform. His method requires the equivalent computational effort of a discrete transform rather than a Fast Fourier Transform. In his process, the amount of computation is the same for all offsets (except zero offset) and for all travel times, whereas it is known that the DMO operator has little effect for small offsets and increasing travel times. The Hale transform cannot exploit that property.
An intractable problem in 3-D DMO processing is that the source-receiver offset is a vector rather than a scaler quantity as it is in 2-D. Furthermore, the data are smeared over an ellipsoidal surface instead of along a line. In a typical 3-D survey where the magnitudes of the offsets and azimuths may vary drastically, the number of separate offsets that must be processed using conventional approaches may approach the number of traces in the survey.